let am, bm, cm be non pair set ; :: thesis: for dm, cin being set
for s being State of (BitFTA3Circ (am,bm,cm,dm,cin))
for a1, a2, a3 being Element of BOOLEAN st a1 = s . am & a2 = s . bm & a3 = s . cm holds
( (Following (s,2)) . (BitFTA3CarryOutput (am,bm,cm,dm,cin)) = 'not' (((('not' a1) '&' ('not' a2)) 'or' (('not' a2) '&' ('not' a3))) 'or' (('not' a3) '&' ('not' a1))) & (Following (s,2)) . (BitFTA3AdderOutputI (am,bm,cm,dm,cin)) = 'not' ((('not' a1) 'xor' ('not' a2)) 'xor' ('not' a3)) )
let dm, cin be set ; :: thesis: for s being State of (BitFTA3Circ (am,bm,cm,dm,cin))
for a1, a2, a3 being Element of BOOLEAN st a1 = s . am & a2 = s . bm & a3 = s . cm holds
( (Following (s,2)) . (BitFTA3CarryOutput (am,bm,cm,dm,cin)) = 'not' (((('not' a1) '&' ('not' a2)) 'or' (('not' a2) '&' ('not' a3))) 'or' (('not' a3) '&' ('not' a1))) & (Following (s,2)) . (BitFTA3AdderOutputI (am,bm,cm,dm,cin)) = 'not' ((('not' a1) 'xor' ('not' a2)) 'xor' ('not' a3)) )
let s be State of (BitFTA3Circ (am,bm,cm,dm,cin)); :: thesis: for a1, a2, a3 being Element of BOOLEAN st a1 = s . am & a2 = s . bm & a3 = s . cm holds
( (Following (s,2)) . (BitFTA3CarryOutput (am,bm,cm,dm,cin)) = 'not' (((('not' a1) '&' ('not' a2)) 'or' (('not' a2) '&' ('not' a3))) 'or' (('not' a3) '&' ('not' a1))) & (Following (s,2)) . (BitFTA3AdderOutputI (am,bm,cm,dm,cin)) = 'not' ((('not' a1) 'xor' ('not' a2)) 'xor' ('not' a3)) )
set S1 = BitGFA3Str (am,bm,cm);
set C1 = BitGFA3Circ (am,bm,cm);
set A1 = GFA3AdderOutput (am,bm,cm);
set A2 = GFA3CarryOutput (am,bm,cm);
set S2 = BitGFA3Str ((GFA3AdderOutput (am,bm,cm)),cin,dm);
set C2 = BitGFA3Circ ((GFA3AdderOutput (am,bm,cm)),cin,dm);
let a1, a2, a3 be Element of BOOLEAN ; :: thesis: ( a1 = s . am & a2 = s . bm & a3 = s . cm implies ( (Following (s,2)) . (BitFTA3CarryOutput (am,bm,cm,dm,cin)) = 'not' (((('not' a1) '&' ('not' a2)) 'or' (('not' a2) '&' ('not' a3))) 'or' (('not' a3) '&' ('not' a1))) & (Following (s,2)) . (BitFTA3AdderOutputI (am,bm,cm,dm,cin)) = 'not' ((('not' a1) 'xor' ('not' a2)) 'xor' ('not' a3)) ) )
assume that
A1: a1 = s . am and
A2: a2 = s . bm and
A3: a3 = s . cm ; :: thesis: ( (Following (s,2)) . (BitFTA3CarryOutput (am,bm,cm,dm,cin)) = 'not' (((('not' a1) '&' ('not' a2)) 'or' (('not' a2) '&' ('not' a3))) 'or' (('not' a3) '&' ('not' a1))) & (Following (s,2)) . (BitFTA3AdderOutputI (am,bm,cm,dm,cin)) = 'not' ((('not' a1) 'xor' ('not' a2)) 'xor' ('not' a3)) )
reconsider s1 = s | the carrier of (BitGFA3Str (am,bm,cm)) as State of (BitGFA3Circ (am,bm,cm)) by FACIRC_1:26;
A4: dom s1 = the carrier of (BitGFA3Str (am,bm,cm)) by CIRCUIT1:3;
am in the carrier of (BitGFA3Str (am,bm,cm)) by GFACIRC1:132;
then A5: a1 = s1 . am by A1, A4, FUNCT_1:47;
reconsider t = s as State of ((BitGFA3Circ (am,bm,cm)) +* (BitGFA3Circ ((GFA3AdderOutput (am,bm,cm)),cin,dm))) ;
A6: InputVertices (BitGFA3Str (am,bm,cm)) misses InnerVertices (BitGFA3Str ((GFA3AdderOutput (am,bm,cm)),cin,dm)) by Lm32;
cm in the carrier of (BitGFA3Str (am,bm,cm)) by GFACIRC1:132;
then A7: a3 = s1 . cm by A3, A4, FUNCT_1:47;
bm in the carrier of (BitGFA3Str (am,bm,cm)) by GFACIRC1:132;
then A8: a2 = s1 . bm by A2, A4, FUNCT_1:47;
GFA3CarryOutput (am,bm,cm) in the carrier of (BitGFA3Str (am,bm,cm)) by GFACIRC1:132;
then (Following (t,2)) . (GFA3CarryOutput (am,bm,cm)) = (Following (s1,2)) . (GFA3CarryOutput (am,bm,cm)) by A6, FACIRC_1:32;
hence (Following (s,2)) . (BitFTA3CarryOutput (am,bm,cm,dm,cin)) = 'not' (((('not' a1) '&' ('not' a2)) 'or' (('not' a2) '&' ('not' a3))) 'or' (('not' a3) '&' ('not' a1))) by A5, A8, A7, GFACIRC1:135; :: thesis: (Following (s,2)) . (BitFTA3AdderOutputI (am,bm,cm,dm,cin)) = 'not' ((('not' a1) 'xor' ('not' a2)) 'xor' ('not' a3))
GFA3AdderOutput (am,bm,cm) in the carrier of (BitGFA3Str (am,bm,cm)) by GFACIRC1:132;
then (Following (t,2)) . (GFA3AdderOutput (am,bm,cm)) = (Following (s1,2)) . (GFA3AdderOutput (am,bm,cm)) by A6, FACIRC_1:32;
hence (Following (s,2)) . (BitFTA3AdderOutputI (am,bm,cm,dm,cin)) = 'not' ((('not' a1) 'xor' ('not' a2)) 'xor' ('not' a3)) by A5, A8, A7, GFACIRC1:135; :: thesis: verum